Vol.:(0123456789) Journal of Mathematical Sciences https://doi.org/10.1007/s10958-022-05899-5 UNIFORM ERGODICITIES OF MARKOV SEMIGROUPS ON ABSTRACT STATES SPACES Nazife Erkurşun‑Özcan 1  · Farrukh Mukhamedov 2,3 Accepted: 22 March 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract The present paper is devoted to the investigation of uniform stabilities of positive C 0 -semigroups defned on abstract state spaces by means of a generalized Dobrushin ergodicity coefcient. The most known results in the literature were obtained for Markov semigroups acting on Banach lattices having unique invariant states. The essence of the present paper is that the considered Markov semigroups (acting on abstract state spaces) do not generally have invariant states and, moreover, abstract state spaces need not necessarily be lattices, implying that the results of the paper are principal in this direction. Keywords Uniform P-ergodicity · C 0 -Markov semigroup · Ergodicity coefcient · Perturbation bound 2010 Mathematics subject classifcation Primary 47A35 · Secondary 60J10 · 28D05. Introduction It is a well-known fact that the asymptotic behavior of certain physical systems constitutes an integral part of physics and mathematics. The solutions of autonomous evolution equations are typically expressed by one-parameter operator semigroups, and therefore, it is essential to know their long-term behavior. Consequently, one needs to have efective methods and techniques for the examination of their asymptotic behavior. In most of the cases, the semigroup acts on some Banach space which is usually taken as certain function space, and thus exhibits some kind of an order structure. In such circumstances, a positive initial value of the evolution equation leads to a positive solution, which allows the consideration of positive operator semigroups. In the literature, diverse ways were discovered to establish, under certain assumptions, the convergence of such semigroups as time goes to infnity [20]. In most cases, the limiting operator is considered to be either compact operator or rank-one projection [2, 9, 11, 19, 25, 26]. This paper is dedicated to the anniversary of Professor S. Samko. * Nazife Erkurşun-Özcan erkursun.ozcan@hacettepe.edu.tr * Farrukh Mukhamedov far75m@yandex.ru; farrukh.m@uaeu.ac.ae 1 Department of Mathematics, Faculty of Science, Hacettepe University, 06800 Ankara, Turkey 2 Department of Mathematical Sciences, College of Science, United Arab Emirates University, 15551 Al-Ain, UAE 3 V.I. Romanovskiy Institute of Mathematics, Uzbek Academy of Sciences, 4, University str, 100125 Tashkent, Uzbekistan